Disclaimer: I know nearly nothing about home repair but am dying to learn and will do the work to ask questions and research. So please be kind...

My situation is that I bought a 1979 split-level house. The steps are the ones without the backing. They're just horizontal steps, you know? The carpet wrapped around them is YUCKY. How could I go about removing the carpet and recarpetting them myself? It doesn't seem like a really difficult thing and I would love some nicer carpet on them...

For this job you would be far better off to get a pro carpet layer to do it. Your not going to have the tools or the know how to make it come out perfect.
Nothing wrong with you removing the old to save money but it's tricky to relay it.

I remember a veteran carpet layer telling me a few years ago (when he was reinstalling new carpet on our stairs that the original installer had messed up a few weeks earlier) that anyone can lay carpet, but getting it right on stairways is where the you can see the difference between amateurs and professionals. Doing it right, getting all of the wrinkles out and hiding all of the edges would be a challenge for anyone.

If you insist on trying it yourself, get a good electric or better yet, pneumatic stapler (you'll also need a compressor for the latter). A good steel straight-edge is essential, as is a quality carpet knife (they're known as widow-makers for a reason, with double exposed sharp edges). Good luck.